`Sign`

definition

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module: IndisputableMonolith.RecogGeom.Examples
domain: RecogGeom
line: 65 · github
papers citing: none yet

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IndisputableMonolith.RecogGeom.Examples on GitHub at line 65.

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All declarations in this module, on Recognition.

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formal source

  62  nonempty := ⟨0⟩
  63
  64/-- Three-valued sign type -/
  65inductive Sign : Type
  66  | neg : Sign
  67  | zero : Sign
  68  | pos : Sign
  69  deriving DecidableEq, Repr
  70
  71/-- Sign is nonempty -/
  72instance : Nonempty Sign := ⟨Sign.zero⟩
  73
  74/-- The sign function -/
  75def signOf : ℤ → Sign
  76  | n => if n < 0 then Sign.neg else if n = 0 then Sign.zero else Sign.pos
  77
  78/-- The sign recognizer on ℤ -/
  79def signRecognizer : Recognizer ℤ Sign where
  80  R := signOf
  81  nontrivial := by
  82    use -1, 1
  83    simp [signOf]
  84
  85/-- **Theorem**: Two integers are indistinguishable iff same sign -/
  86theorem sign_indistinguishable (n m : ℤ) :
  87    Indistinguishable signRecognizer n m ↔ signOf n = signOf m := by
  88  rfl
  89
  90/-- **Theorem**: -5 ~ -3 (both negative) -/
  91theorem neg_indist : Indistinguishable signRecognizer (-5) (-3) := by
  92  simp [Indistinguishable, signRecognizer, signOf]
  93
  94/-- **Theorem**: -1 ≁ 1 (different signs) -/
  95theorem neg_pos_dist : ¬Indistinguishable signRecognizer (-1) 1 := by

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